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State-constrained optimal control of the continuity equation and infinite-dimensional viscosity solutions

Fabio Bagagiolo, Ivan Romanò

Abstract

We study a finite horizon optimal control problem for the continuity equation under a weighted integral state constraint on the mass outside a fixed set. The model is cast in a Hilbert framework for densities. On a suitable invariant compact subset, we prove that the value function is Lipschitz continuous and satisfies, by dynamic programming, the associated infinite dimensional constrained Hamilton Jacobi Bellman equation in viscosity sense (subsolution in the interior, supersolution up to the boundary). We finally prove a comparison principle and uniqueness in the Lipschitz class.

State-constrained optimal control of the continuity equation and infinite-dimensional viscosity solutions

Abstract

We study a finite horizon optimal control problem for the continuity equation under a weighted integral state constraint on the mass outside a fixed set. The model is cast in a Hilbert framework for densities. On a suitable invariant compact subset, we prove that the value function is Lipschitz continuous and satisfies, by dynamic programming, the associated infinite dimensional constrained Hamilton Jacobi Bellman equation in viscosity sense (subsolution in the interior, supersolution up to the boundary). We finally prove a comparison principle and uniqueness in the Lipschitz class.

Paper Structure

This paper contains 10 sections, 14 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. The continuity equation
  3. Regularity of the solutions
  4. An invariant and compact set for the evolutions
  5. The state-constrained optimal control problem
  6. Value function regularity
  7. The constrained Hamilton--Jacobi--Bellman equation
  8. Comparison principle and uniqueness
  9. Some comments about different formulations of the state constraint
  10. Appendix

Key Result

Proposition 2.3

There exists a constant $C=C(M,T,n)$ such that, for every $t\in[0,T)$, every $\bar{m}\in W^{2,\infty}(\mathbb{R}^n)$, and every $\alpha\in\mathcal{A}$, one has An analogous estimate also holds for the $H^2$ norm, starting from an initial datum in $H^2(\mathbb{R}^n)$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (35)

  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Remark 2.7
  • Proposition 2.8
  • proof
  • ...and 25 more